Difference between revisions of "2019 AMC 12B Problems/Problem 12"
Magentacobra (talk | contribs) (→Solution 1.5 (Little bit of coordinate bash)) |
m (→Solution 1.5 (Little bit of coordinate bash)) |
||
Line 51: | Line 51: | ||
~Solution by MagentaCobra | ~Solution by MagentaCobra | ||
+ | |||
+ | ==Solution 0.1== | ||
+ | One answer choice stands out above the rest, and that is <math>\boxed{\textbf{(D)} = \frac{7}{9}}</math>. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2019|ab=B|num-b=11|num-a=13}} | {{AMC12 box|year=2019|ab=B|num-b=11|num-a=13}} |
Revision as of 20:29, 14 February 2019
Contents
Problem
Right triangle with right angle at is constructed outwards on the hypotenuse of isosceles right triangle with leg length , as shown, so that the two triangles have equal perimeters. What is ?
Solution 1
Observe that the "equal perimeter" part implies that . A quick Pythagorean chase gives . Use the sine addition formula on angles and (which requires finding their cosines as well), and this gives the sine of . Now, use on angle to get .
Feel free to elaborate if necessary.
Solution 1.5 (Little bit of coordinate bash)
After using Pythagorean to find and , we can instead notice that the angle between the y-coordinate and is degrees, and implies that the slope of that line is 1. If we draw a perpendicular from point , we can then proceed to find the height and base of this new triangle (defined by where is the intersection of the altitude and ) by coordinate-bashing, which turns out to be and respectively.
By double angle formula and difference of squares, it's easy to see that our answer is
~Solution by MagentaCobra
Solution 0.1
One answer choice stands out above the rest, and that is .
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |